3.26 \(\int \frac{x^5}{x-x^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{x^3}{3}-x+\tanh ^{-1}(x) \]

[Out]

-x - x^3/3 + ArcTanh[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0229495, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{x^3}{3}-x+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(x - x^3),x]

[Out]

-x - x^3/3 + ArcTanh[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.68189, size = 8, normalized size = 0.62 \[ - \frac{x^{3}}{3} - x + \operatorname{atanh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**3+x),x)

[Out]

-x**3/3 - x + atanh(x)

_______________________________________________________________________________________

Mathematica [B]  time = 0.00682492, size = 29, normalized size = 2.23 \[ -\frac{x^3}{3}-x-\frac{1}{2} \log (1-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(x - x^3),x]

[Out]

-x - x^3/3 - Log[1 - x]/2 + Log[1 + x]/2

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 22, normalized size = 1.7 \[ -{\frac{{x}^{3}}{3}}-x-{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^3+x),x)

[Out]

-1/3*x^3-x-1/2*ln(-1+x)+1/2*ln(1+x)

_______________________________________________________________________________________

Maxima [A]  time = 1.3692, size = 28, normalized size = 2.15 \[ -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^3 - x),x, algorithm="maxima")

[Out]

-1/3*x^3 - x + 1/2*log(x + 1) - 1/2*log(x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.204217, size = 28, normalized size = 2.15 \[ -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^3 - x),x, algorithm="fricas")

[Out]

-1/3*x^3 - x + 1/2*log(x + 1) - 1/2*log(x - 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.170204, size = 19, normalized size = 1.46 \[ - \frac{x^{3}}{3} - x - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**3+x),x)

[Out]

-x**3/3 - x - log(x - 1)/2 + log(x + 1)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219067, size = 31, normalized size = 2.38 \[ -\frac{1}{3} \, x^{3} - x + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/(x^3 - x),x, algorithm="giac")

[Out]

-1/3*x^3 - x + 1/2*ln(abs(x + 1)) - 1/2*ln(abs(x - 1))